Group #30662172 — subgroups

Subgroups — #1989×C₁₃₁ #30662172

Order 16768.

260 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#125×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#340×C₁₃₁	C₂	1	2	yes	maximal
5	#340×C₁₃₁	C₂	1	2	yes	maximal
6	C₂²×C₈×C₁₃₁	C₂²	1	2	yes	—
7	#125×C₁₃₁	—	2	2	no	—
8	#125×C₁₃₁	C₄	1	4	yes	complemented
9	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
10	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
11	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₃₁	—	2	1	no	—
15	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₃₁	—	2	2	no	—
17	#95×C₁₃₁	—	2	1	no	—
18	C₂²×C₄×C₁₃₁	—	2	2	no	—
19	C₂²×C₄×C₁₃₁	—	2	1	no	—
20	C₂²×C₄×C₁₃₁	—	2	1	no	—
21	C₂²×C₄×C₁₃₁	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₃₁	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₁₃₁	—	2	2	no	—
25	C₂×C₈×C₁₃₁	—	2	4	no	—
26	C₄⋊C₄×C₁₃₁	—	2	4	no	—
27	C₄⋊C₄×C₁₃₁	—	2	4	no	—
28	C₄⋊C₄×C₁₃₁	—	2	1	no	—
29	C₄⋊C₄×C₁₃₁	—	2	1	no	—
30	C₄⋊C₄×C₁₃₁	—	2	2	no	—
31	C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
32	C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
33	C₄⋊C₄×C₁₃₁	Q₈	1	4	yes	—
34	C₄²×C₁₃₁	—	4	2	no	—
35	C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
36	C₄²×C₁₃₁	D₄	1	2	yes	—
37	C₂³×C₁₃₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₃₁	—	2	2	no	—
39	C₂×C₄×C₁₃₁	—	2	1	no	—
40	C₂×C₄×C₁₃₁	—	2	1	no	—
41	C₂×C₄×C₁₃₁	—	2	2	no	—
42	C₂×C₄×C₁₃₁	—	2	2	no	—
43	C₂×C₄×C₁₃₁	—	2	4	no	—
44	C₂×C₄×C₁₃₁	—	4	2	no	—
45	C₂×C₄×C₁₃₁	—	4	2	no	—
46	C₂×C₄×C₁₃₁	—	4	2	no	—
47	C₂×C₄×C₁₃₁	—	4	1	no	—
48	C₂×C₄×C₁₃₁	—	4	1	no	—
49	C₂×C₄×C₁₃₁	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₁₃₁	C₂²⋊C₄	1	2	yes	—
51	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₃₁	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₁₃₁	#41	1	2	yes	—
55	C₈×C₁₃₁	—	4	2	no	—
56	C₂²×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₃₁	C₂²⋊Q₈	1	2	yes	—
58	C₂²×C₁₃₁	#135	1	2	yes	—
59	C₂²×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂²×C₁₃₁	#137	1	1	yes	characteristic
61	C₄×C₁₃₁	—	2	2	no	—
62	C₄×C₁₃₁	—	2	2	no	—
63	C₄×C₁₃₁	—	4	2	no	—
64	C₄×C₁₃₁	—	4	2	no	—
65	C₄×C₁₃₁	—	4	4	no	—
66	C₄×C₁₃₁	C₄×Q₈	1	2	yes	—
67	C₄×C₁₃₁	#124	1	1	yes	characteristic
68	C₄×C₁₃₁	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₁₃₁	#391	1	1	yes	characteristic
70	C₂×C₁₃₁	#416	1	1	yes	characteristic
71	C₂×C₁₃₁	#417	1	1	yes	characteristic
72	C₂×C₁₃₁	#478	1	2	yes	—
73	C₂×C₁₃₁	#479	1	2	yes	—
74	C₁₃₁	#1989	1	1	yes	characteristic, direct product factor, Sylow
75	#1989	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂×#125	C₂×C₁₃₁	1	1	yes	characteristic
77	C₂×C₄⋊C₈	C₂×C₁₃₁	1	1	yes	characteristic
78	C₄×C₄⋊C₄	C₂×C₁₃₁	1	1	yes	characteristic
79	#340	C₂×C₁₃₁	1	2	yes	—
80	#340	C₂×C₁₃₁	1	2	yes	—
81	C₂²×C₈	C₂²×C₁₃₁	1	2	yes	—
82	#125	—	2	2	no	—
83	#125	C₄×C₁₃₁	1	4	yes	complemented
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	2	1	no	—
86	C₂×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
87	C₂×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
88	C₂×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
89	C₂×C₄²	—	2	1	no	—
90	C₂×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
91	C₄⋊C₈	—	2	2	no	—
92	#95	—	2	1	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	D₄×C₁₃₁	1	1	yes	characteristic
97	C₂²×C₄	D₄×C₁₃₁	1	1	yes	characteristic
98	C₂²×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	4	no	—
101	C₄⋊C₄	—	2	4	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	C₂×C₄×C₁₃₁	1	2	yes	—
107	C₄⋊C₄	C₂×C₄×C₁₃₁	1	2	yes	—
108	C₄⋊C₄	Q₈×C₁₃₁	1	4	yes	—
109	C₄²	—	4	2	no	—
110	C₄²	C₂×C₄×C₁₃₁	1	2	yes	—
111	C₄²	D₄×C₁₃₁	1	2	yes	—
112	C₂³	C₂×D₄×C₁₃₁	1	1	yes	characteristic
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	C₂²⋊C₄×C₁₃₁	1	2	yes	—
125	C₂×C₄	C₂²⋊C₄×C₁₃₁	1	2	yes	—
126	C₂×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
127	C₂×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
128	C₂×C₄	C₂×Q₈×C₁₃₁	1	2	yes	—
129	C₂×C₄	#41×C₁₃₁	1	2	yes	—
130	C₈	—	4	2	no	—
131	C₂²	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
132	C₂²	C₂²⋊Q₈×C₁₃₁	1	2	yes	—
133	C₂²	#135×C₁₃₁	1	2	yes	—
134	C₂²	AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
135	C₂²	#137×C₁₃₁	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	4	2	no	—
139	C₄	—	4	2	no	—
140	C₄	—	4	4	no	—
141	C₄	C₄×Q₈×C₁₃₁	1	2	yes	—
142	C₄	#124×C₁₃₁	1	1	yes	characteristic
143	C₄	C₄⋊Q₈×C₁₃₁	1	1	yes	characteristic
144	C₂	#391×C₁₃₁	1	1	yes	characteristic
145	C₂	#416×C₁₃₁	1	1	yes	characteristic
146	C₂	#417×C₁₃₁	1	1	yes	characteristic
147	C₂	#478×C₁₃₁	1	2	yes	—
148	C₂	#479×C₁₃₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#125×C₁₃₁

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₁₃₁

C₂

yes

maximal, characteristic

C₄×C₄⋊C₄×C₁₃₁

C₂

yes

maximal, characteristic

#340×C₁₃₁

C₂

yes

maximal

#340×C₁₃₁

C₂

yes

maximal

C₂²×C₈×C₁₃₁

C₂²

yes

—

#125×C₁₃₁

—

#125×C₁₃₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₃₁

—

C₂×C₄⋊C₄×C₁₃₁

—

C₂×C₄⋊C₄×C₁₃₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₃₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₃₁

C₂²

yes

—

C₂×C₄²×C₁₃₁

—

C₂×C₄²×C₁₃₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₁

—

#95×C₁₃₁

—

C₂²×C₄×C₁₃₁

—

C₂²×C₄×C₁₃₁

—

C₂²×C₄×C₁₃₁

—

C₂²×C₄×C₁₃₁

D₄

yes

characteristic

C₂²×C₄×C₁₃₁

D₄

yes

characteristic

C₂²×C₄×C₁₃₁

C₂³

yes

characteristic

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

C₂×C₄

yes

—

C₄⋊C₄×C₁₃₁

C₂×C₄

yes

—

C₄⋊C₄×C₁₃₁

Q₈

yes

—

C₄²×C₁₃₁

—

C₄²×C₁₃₁

C₂×C₄

yes

—

C₄²×C₁₃₁

D₄

yes

—

C₂³×C₁₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂×Q₈

yes

—

C₂×C₄×C₁₃₁

#41

yes

—

C₈×C₁₃₁

—

C₂²×C₁₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃₁

C₂²⋊Q₈

yes

—

C₂²×C₁₃₁

#135

yes

—

C₂²×C₁₃₁

yes

C₂²×C₁₃₁

yes

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

C₄×Q₈

yes

—

C₄×C₁₃₁

#124

yes

characteristic

C₄×C₁₃₁

C₄⋊Q₈

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

—

C₂×C₁₃₁

#479

yes

—

C₁₃₁

#1989

yes

characteristic, direct product factor, Sylow

#1989

C₁₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×#125

C₂×C₁₃₁

yes

characteristic

C₂×C₄⋊C₈

C₂×C₁₃₁

yes

characteristic

C₄×C₄⋊C₄

C₂×C₁₃₁

yes

characteristic

#340

C₂×C₁₃₁

yes

—

#340

C₂×C₁₃₁

yes

—

C₂²×C₈

C₂²×C₁₃₁

yes

—

#125

—

#125

C₄×C₁₃₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₁₃₁

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₁₃₁

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₁₃₁

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×C₁₃₁

yes

characteristic

C₄⋊C₈

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₁₃₁

yes

characteristic

C₂²×C₄

D₄×C₁₃₁

yes

characteristic

C₂²×C₄

C₂³×C₁₃₁

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

C₂×C₄×C₁₃₁

yes

—

107

C₄⋊C₄

C₂×C₄×C₁₃₁

yes

—

108

C₄⋊C₄

Q₈×C₁₃₁

yes

—

109

C₄²

—

110

C₄²

C₂×C₄×C₁₃₁

yes

—

111

C₄²

D₄×C₁₃₁

yes

—

112

C₂³

C₂×D₄×C₁₃₁

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

C₂²⋊C₄×C₁₃₁

yes

—

125

C₂×C₄

C₂²⋊C₄×C₁₃₁

yes

—

126

C₂×C₄

C₂²×C₄×C₁₃₁

yes

characteristic

127

C₂×C₄

C₂×D₄×C₁₃₁

yes

characteristic

128

C₂×C₄

C₂×Q₈×C₁₃₁

yes

—

129

C₂×C₄

#41×C₁₃₁

yes

—

130

C₈

—

131

C₂²

C₂×C₂²⋊C₄×C₁₃₁

yes

characteristic

132

C₂²

C₂²⋊Q₈×C₁₃₁

yes

—

133

C₂²

#135×C₁₃₁

yes

—

134

C₂²

AGL(1,ℤ/8)×C₁₃₁

yes

characteristic

135

C₂²

#137×C₁₃₁

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

C₄×Q₈×C₁₃₁

yes

—

142

C₄

#124×C₁₃₁

yes

characteristic

143

C₄

C₄⋊Q₈×C₁₃₁

yes

characteristic

144

C₂

#391×C₁₃₁

yes

characteristic

145

C₂

#416×C₁₃₁

yes

characteristic

146

C₂

#417×C₁₃₁

yes

characteristic

147

C₂

#478×C₁₃₁

yes

—

148

C₂

#479×C₁₃₁

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₄	C₂³×C₁₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₁₃₁

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₁₃₁

Frattini, cosocle

C₂²×C₄

C₂³×C₁₃₁

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ G

N	H ≅ G/N	type
8. #125×C₁₃₁	C₄	semidirect
74. C₁₃₁	#1989	direct (N × H)
75. #1989	C₁₃₁	direct (N × H)
83. #125	C₄×C₁₃₁	semidirect

H ≅ G/N

type

8. #125×C₁₃₁

C₄

semidirect

74. C₁₃₁

#1989

direct (N × H)

75. #1989

C₁₃₁

direct (N × H)

83. #125

C₄×C₁₃₁

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#125×C₁₃₁	2
C₂×C₄⋊C₈×C₁₃₁	2
C₄×C₄⋊C₄×C₁₃₁	2
#340×C₁₃₁	2
#1989	131

subgroup

index

C₂×#125×C₁₃₁

C₂×C₄⋊C₈×C₁₃₁

C₄×C₄⋊C₄×C₁₃₁

#340×C₁₃₁

#1989

131

Subgroups — #1989×C₁₃₁ #30662172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1989×C131 #30662172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1989×C₁₃₁ #30662172